If somebody has permission to edit my eGC1, does that mean they can also view and edit a budget connected to that eGC1?

Not necessarily. A person’s ability to edit a connected budget is determined by the access they are granted to that budget. If a person with access to an eGC1 has not explicitly been assigned access to a connected budget, they will be able to open and review the budget from the eGC1, but they will not be able to edit it.